AP Calculus AB and BC glossary

Power function

A power function is a constant times x raised to a fixed real exponent, such as 3x cubed, the square root of x, or 1 over x squared. The variable is the base and the exponent is a constant, which is exactly the setup the power rule differentiates.

Formally $f(x) = k x^{n}$ with $k$ and $n$ constant. The exponent may be any real number, whole, negative, fractional, or irrational, and the single rule $\frac{d}{dx}\left[k x^{n}\right] = k n x^{n-1}$ handles every one of them. The work on a free response question is almost never the differentiating: it is the rewriting that comes first.

Written as	Power form	Derivative
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\frac{1}{x^{3}}$	$x^{-3}$	$-3x^{-4}$
$\sqrt[3]{x^{2}}$	$x^{2/3}$	$\frac{2}{3}x^{-1/3}$
$\frac{x^{2}+1}{x}$	$x + x^{-1}$	$1 - x^{-2}$

The mistake

Reading $2^{x}$ as a power function. Its variable sits in the exponent, so the power rule does not apply and the derivative is $2^{x} \ln 2$ , never $x \cdot 2^{x-1}$ . The test is where the variable lives: in the base it is a power function, in the exponent it is exponential, and in both at once, as in $x^{x}$ , it is neither and needs logarithmic differentiation.

Appears in: Unit 2: Defining the Derivative